Sir Ken Morrison, the no-nonsense Yorkshireman who built up the supermarket chain from his father's market stalls, famously dismissed home deliveries as "something he had done on his bike as a young man".

With Sir Ken no longer in charge, Morrisons is less sniffy about home delivery – but it remains the only big UK supermarket to shun the booming business of online groceries, which analysts expect to be worth£7.5bn this year.

Investors are getting impatient with Morrisons' wariness. Dalton Philips, who came in as chief executive after Marc Bolland jumped to M&S, has said the online grocery business "is something that we are going to have to do in the coming months".

He is under mounting pressure to explain his plans as he prepares to announce the company's full-year results next Thursday. Morrisons' internet strategy could make or break the Irish chief executive, who marks his third anniversary in the top job this month.

"They have got to come out definitively and say what they are doing," said Neil Saunders, managing director of Conlumino retail analysts. "The market is looking for a commitment or otherwise to do online, with a time frame on that, and they are looking for some detail on how the offer will be constructed."

Morrisons, which had the worst performance of any supermarket chain over the lucrative Christmas period, risks being left behind as the online market soars, analysts warn. When it goes online, it will have to run fast to catch up with rivals who have been selling food over the internet for more than a decade. Tesco.com went live in 2000. Ocado, the upmarket online grocer that sells Waitrose goods, dispatched its first van in 2002.

"Online groceries is about 4.5% to 5% of the market. It is growing around 15% per annum and by 2020 we think it will be worth between 10-12% of the market," said Clive Black, head of research at Shore Capital. "[By 2020] Morrisons would effectively be competing only with 88% of the market and that is a very difficult task."

Morrisons blamed its miserable Christmas trading partly on the absence of an online grocery business – like for like sales were down 2.5% in the last six weeks of the year.

It has been taking tentative steps in cyberspace. In 2011 it bought Kiddicare.com. Last year it launched wine website Morrisons Cellar, where a bottle starts at £3.99 and customers can do a taste test devised by posh north London wine merchant Bibendum. This spring, the retailer will launch a kitchen products website with home-shopping specialist Lakeland, bringing its tally of online businesses to three.

And in a move to become more internet savvy, in 2011 Morrisons acquired a 10% stake in Fresh Direct, an online grocer delivering farm-fresh produce to well-heeled New Yorkers.

Morrisons snapped up 56 stores from bust retailers Blockbuster Video and HMV last month, as it seeks to increase its portfolio of convenience stores – another gap in its business that gives investors a headache.

Although Philips agrees the online grocery business cannot be ignored, he strikes a wary note. The "genius of the supermarket", he said recently, is that customers carry the costs of picking out the goods and shuttling to and from the shop. If supermarkets take this bit over, prices could rise for those shoppers who prefer the bricks-and-mortar experience.

"You can end up therefore subsidising your online shopper by charging your core customer more and that is something that philosophically I struggle with," he told BBC You and Yours.

Bill Grimsey, former chief executive of Wickes, Iceland and Focus DIY, says Morrisons cannot afford to stay offline as consumers abandon bricks-and-mortar shops. "Do Morrisons have a choice to stay out of the online food shopping business? The answer is unequivocally no. If they don't, Tesco, Sainsbury's and Asda will be taking shoppers [Morrisons] could have earned."

Stanislav Desyatnikov, 42, a software engineer, might be just the customer Morrisons is missing out on. Popping into Morrisons in north London to pick up some juice, he said he and his wife shop online for groceries once or twice a month – at Tesco. If Morrisons offered home shopping, he could be tempted. "I work nearby, so I could have the food delivered to my office."

When Morrisons takes the plunge, it must decide whether to ship orders out from the network of 455 supermarkets, or use "dark stores", drab dotcom-only warehouses that are springing up on industrial estates across the country to feed the hunger for online shopping.

Analysts expect Morrisons will choose store delivery to get their fresh produce speedily to customers. "Morrisons has a great advantage in that it really is top of the pack for fresh food," said Andrew Stevens, senior retail analyst at Verdict. A 'fulfilment from store' strategy would be the best way to translate these fresh-food values, he said.

"Though in London and the south-east, it will certainly resort to warehouses, because it has fewer stores."

This combination of store delivery and warehouses in the south east is favoured by Tesco and Asda, while Sainsbury's has traditionally preferred delivery from its shops.

But Morrisons will have to open its wallet if it wants to elbow its way into this crowded market. It could try to tempt customers away from rivals with free delivery, fistfuls of discount coupons, or the increasingly popular "click and collect" service that allows customers to pick up their online orders in-store

Speculation persists that Morrisons could make a bid for Ocado. This would give them "a leg up in the market", said Saunders of Conlumino, "but they would have to think seriously because Ocado is not a profit-making operation at the moment".

Not everyone is convinced internet shopping is the answer to Morrisons' recent sales slowdown. Some analysts accuse the retailer of deserting its core customers, by moving up-market with its "Fresh Format" revamp, stores offering treats such as artisanal bread and olive counters. "Fresh Format is visually lovely," said Clive Black at Shore Capital. "But we think they have disenfranchised some of their traditional customers and failed to attract new ones," he said. "The fresh herbs and the cupcakes haven't attracted the customers Morrisons seem to have been aspiring too."

Philips has always stoutly rejected the charge of losing touch with his customers. He also insists being last to online shopping is no bad thing – because Morrisons can learn from rivals' mistakes.

But time is running out. "There is no last-mover advantage. It is a fiction," said Saunders. "The problem is that the grocery market is saturated.

"Morrisons has now got to steal the show from every other player in order to win business."

Other online refuseniks:

Aldi and Lidl: Tight times mean "hard discounters" Aldi and Lidl are booming – without a single sale online. With "a lean and efficient logistical model" that eschews "luxuries, such as customer service and plastic bags" it is hard to see why they would go for the "frippery" of a pricey online service for groceries, said Bryan Roberts at Kantar retail consultancy. But online shopping could make sense for their non-food products, he added.

Poundland: Everything for a pound might not sound like a promising start for a home shopping service, but Poundland could make it work. Rival discounter Poundworld created its Discount Wholesale website, which sells everything from novelty ties to pregnancy tests. Targeted at small traders, Discount Wholesale demands a minimum spend of £500.

Primark: Booming sales show being offline isn't hurting, but supply chain director Martin White has said it would be "dumb as a business if we weren't looking online". Clive Black, head of research at Shore Capital, thinks it unlikely Primark will go online soon, but it will "develop its website so it is more of a showroom".